NZ Level 7 (NZC) Level 2 (NCEA) Area of Circles and Sectors
Lesson

We already know that area is the space inside a 2D shape.  We can find the area of a circle, but we will need a special rule.

The following investigation will demonstrate what happens when we unravel segments of a circle.

Interesting isn't it that when we realign the segments we end up with a parallelogram shape.  Which is great, because it means we know how to find the area based on our knowledge that the area of a parallelogram has formula $A=bh$A=bh.  In a circle, the base is half the circumference and the height is the radius.

Area of a Circle

$\text{Area of a circle}=\pi r^2$Area of a circle=πr2

### What if we don't have an entire circle?

Well, half a circle would have half the area or half the circumference. One quarter of a circle would have a quarter of the area, or a quarter of the circumference.  In fact all we need to know is what fraction the sector is of a whole circle.  For this all we need to know is the angle of the sector.

Looking at the quarter circle, the angle of the sector is $90$90°. The fraction of the circle is $\frac{90}{360}=\frac{1}{4}$90360=14

More generally, If the angle of the sector is $\theta$θ, then the fraction of the circle is represented by

$fraction=\frac{\theta}{360}$fraction=θ360 (due to there being $360$360° in a circle).

#### Example

Question: Find the area of a sector with central angle of $126$126° and radius of $7$7cm. Evaluate to $2$2 decimal places.

Think: What fraction is this sector of a whole circle?  What is the rule for area?

Do:  This sector is $\frac{126}{360}=0.35$126360=0.35 of a circle. Area of a circle is $A=\pi r^2$A=πr2, so the area of the sector is

 $0.35\times\pi r^2$0.35×πr2 $=$= $0.35\pi\times7^2$0.35π×72 $=$= $17.15\pi$17.15π $=$= $53.88$53.88 cm2  (rounded to 2 decimal places)

#### More Worked Examples

##### Question 1

If the radius of the circle is $5$5 cm, find its area.

##### Question 2

Find the area of the shaded region in the following figure, correct to 1 decimal place. ##### Question 3

Find the area of the shaded region in the following figure, correct to 1 decimal place. ##### Question 4

Consider the sector below. 1. Calculate the perimeter. Give your answer correct to $2$2 decimal places.

2. Calculate the area. Give your answer correct to $2$2 decimal places.

##### Question 5

Consider the sector below. ##### QUESTION 6

A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over? ### Outcomes

#### M7-4

Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions

#### 91259

Apply trigonometric relationships in solving problems